Massachusetts Institute of Technology
Department of Physics
Physics 8.022
Fall 2002
Special Relativity
Peter Fisher
Abstract
This write-up summarizes the derivation and use of the Lorentz
transformation to relate two inertial frames. The main results are
Lorentz length contraction and the time dilation. The derivation of
the way of adding velocities is also given.
Inertial Frames and Einstein’s Postulate
Inertial frames are frames related by a constant velocity along any axis. An
object is at rest in an inertial frame when its position in space does not
change with time. We consider two inertial frames: the 26-100 (S) frame in
which we are at rest and the Space Shuttle frame (S ) which moves relative
to S with a velocity u along the x axis and in which the Shuttle is at rest.
Newton postulated that the laws of physics are the same in all inertial frames;
specific predictions made in one frame must also be true in all other inertial
frames.
Based on the result of the Michelson-Morley experiment, Einstein postulated:
There is a universal constant, c, which is the same in all inertial frames. This
velocity corresponds to the velocity of light, c = 3 × 1010 cm/s. We will now
investigate the implications of Einstein’s postulate.
Galilean Transformation
Let the coordinates (x, t) denote the location of an event in the 26-100 frame
and (x , t ) denote the location of the same event in the Shuttle frame. The
Galilean transformation which relates the two coordinate systems is:
x
y
z
t
x − ut
y
z
t
=
=
=
=
(1)
which tells us we just need to take into account the relative translation of the
coordinate systems to transform the coordinates of a given event from one
frame to another. Suppose a body moves with velocity v inside the Shuttle.
What velocity v do we in 26-100 see it move with? Using the transformation
Eq. 1 from Shuttle coordinates to 26-100 coordinates gives:
x = x + ut
→ dx = dx + udt
dx
dx
dt
= v = + u = v + u
→
dt
dt
dt
This just says we in 26-100 see the body moving with the velocity v it has
in the shuttle frame plus the velocity of the Shuttle frame (u) with respect
to 26-100. If Einstein’s postulate that the velocity of light is the same in all
inertial frames (based on the result of the Michelson-Morley experiment) is
correct, then the Galilean transformation is not the correct way to transform
space-time coordinates between two inertial frames.
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Derivation of the Lorentz Transformation
We wish to find a transformation between the 26-100 frame (S) and the
Shuttle frame (S ) which such that the velocity of light c is the same in both
coordinate systems. The most general linear transformation (i.e. depending
on x and t , but not higher powers like x2 , etc.) 1 is:
x
y
z
t
=
=
=
=
Ax + Bt
y
z
Dx + Et
(2)
(3)
remember the velocity u is along the x-axis, so the other coordinates must
remain the same: y = y, z = z. We have chosen at x = x = 0 at t = t = 0.
The origin of the S frame (x = 0) moves along the trajectory x = −ut in
the S frame. Putting this into the Eq. 2 above gives
0 = Aut + Bt → B = −Au → x = A(x − ut).
In the frame S the origin (x = 0) of S describes a trajectory along x = ut
and putting this into equation Eq. 2 above gives −ut = A(x − ut).
Substituting this result into Eq. 3 gives −u(Dx + Et) = A(x − ut) and x = 0
implies −uEt = −Aut → E = A. Now we have
x = A(x − ut)
t = Dx + At.
(4)
(5)
Next, we apply Einstein’s Postulate: The velocity of light is the same in all
inertial frames. A pulse of light emitted at t = 0 along the x axis in the S frame follows the trajectory x = ct . We use this to relate Eqs. 4 and 5:
A(x − ut) = c(Dx + At).
1
(6)
It can be shown that transformations involving terms like x2 lead to transformations
which are not invariant for translations of the coordinate systems.
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The same pulse of light would be seen as following the trajectory x = ct in
the S frame. Use this to eliminate x in Eq. 6: A(c − u)t = c(Dc + A)t →
D = −Au/c2 and we have
x = A(x − ut)
ux
t = A(− 2 + t).
c
(7)
(8)
Finally, what happens to a light pulse emitted along the y axis at time t = 0
in frame S? It follows a trajectory y = ct and x = 0 in frame S and a
trajectory x2 + y 2 = ct2 in frame S . If we substitute for x , t using Eq. 7
and 8 and y = y = ct , we have:
A2 (x − ut)2 + c2 t2 = c2 A2 (−
and x = 0 implies
ux
+ t)
c2
1
.
A= �
1 − u2 /c2
√
Since this expression occurs so often, pros use γ = 1/ 1 − β 2 where β = u/c.
The positive value of the square root corresponds to the physical solution and
we have the final form of the Lorentz transformation:
1
(x − ut)
x = �
1 − u2 /c2
y = y
z = z
(9)
(10)
(11)
1
ux
t = �
(t − 2 ).
c
1 − u2 /c2
Going from frame S to S, we will have
1
x = �
(x + ut )
2
2
1 − u /c
1
ux
t = �
(t + 2 ).
c
1 − u2 /c2
4
(12)
Simple examples of the Lorentz Transformations
Particle at rest or moving at constant velocity
Fig. 1 shows the coordinate system in the 26-100 frame. A particle at rest
at the point xo follows a trajectory in a coordinate system (called a world
line) as shown in the figure. An event represents an occurrence at a specific
coordinate and time, for example at (xo , to ) in the figure. Suppose a particle
moves with velocity u along the x axis in 26-100 (frame S). Fig. 2 shows
the world line for a particle moving in 26-100 frame with velocity u passing
through (xo , to ). Now suppose a particle is at rest in the Shuttle (frame S )
at coordinate x = 0 . To the people in the shuttle, the coordinate system
appears as in Fig. 3.
We put primes to indicate this is the rest frame in the Shuttle. A particle at
rest at xo in the Shuttle describes the world line shown.
Coordinate systems
What does the Shuttle coordinate system look like to us in 26-100 if we
arrange for the shuttle to go by in such a way that at t = 0, t = x = x = 0?
The x axis is all those points where t = 0, so to see what the x axis looks
like in the 26-100 frame, put t = 0 into the Lorentz transformations Eq. 12:
1
x = �
x
2
2
1 − u /c
ux
1
.
t = �
1 − u2 /c2 c
Eliminating x gives x = c2 t/u which is the equation of the x axis in the
26-100 coordinate system. The t axis is the locus of all points such that
x = 0 and putting x = 0 into the Lorentz equations and eliminating t gives
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t
World line
to
Event
xo
x
Figure 1: Coordinate system for the 26-100 frame.
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t
slope=1/u
(xo,yo)
x
Figure 2: 26-100 showing the world line for a particle moving with velocity
u passing through (xo , to ).
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t’
World line
(x’o,t’o)
Event
x’o
x’
Figure 3: World line for a particle at rest with respect to the Shuttle. Both
move with velocity u in the 26-100 frame.
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t
t’
Slope=1/u
Event a
ta
Slope=u/c2
t’a
x’a
xa
x’
x
Figure 4: Coordinates of Event a in both S and S frames.
t = x/u for the equation of the t axis in the 26-100 rest frame, shown in
Fig. 4
The event a is located at a unique point in space-time which is described by
(xa , ta ) in the 26-100 frame and (xa , ta ) in the Shuttle frame. The coordinates
(xa , ta ) and (xa , ta ) are related by the Lorentz transformation. Be sure you
can answer the following questions:
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• What happens to the x and t axis as viewed from the 26-100 frame as
u goes to zero?
• What happens to the x and t axis as viewed from the 26-100 frame as
u goes to c, the velocity of light?
The Lorentz Contraction
Suppose there is a 2×4 2 located of length L located in the Shuttle with the
left end located at x = 0 and the right end located at x = L. At time t = 0,
the Shuttle Mission Specialist in Wood Measuring measures the length of the
2×4. He is aided by a light on each end of the 2×4 which flash at the time
of the measurement, t = 0. The lights define two events, one at (0, 0) and
the other at (L, 0) in the Shuttle rest frame. The measurement events are
shown in Fig. 5. We define measuring the length as the difference in spatial
coordinates of two simultaneous events.
In the 26-100 frame, we see the Shuttle come roaring through 26-100 at
velocity u and at t = t = 0 the origin of the Shuttle coordinate system
corresponds to the origin of the 26-100 coordinate system. We can use the
Lorentz transformation to determine the coordinates of the measurement
events on board the Shuttle. The left end measurement event is (0, 0) in
the Shuttle frame and putting into the Lorentz transformation gives (0, 0) in
the 26-100 frame as well. The right end measurement event is (L, 0) in the
Shuttle frame and the Lorentz transformation to the 26-100 frame gives
L
uL
(�
,�
).
1 − u2 /c2
1 − u2/c2
Notice that it appears to us in the 26-100 frame that the measurement of the
right end appears to have been made later than the measurement of the left
end while on the Shuttle they were both made at t = 0. Diagrammatically,
the situation in the 26-100 frame appears as in Fig. 6.
In the 26-100 frame, we measure the length of the 2×4 at t = 0 . The left end
coincides with the origin at t = 0 . The right end follows along a world line
2
A “2×4” is a US standard piece of wood 1 1/2 by 3 1/2 inches and several feet long.
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t’
right
world line
right
world line
Measurement
events,
Shuttle
Event
x’o
x’
L
Figure 5: Measurement of 2×4 in the 26-100 frame.
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t
left
world line
right
world line
t’
Measurement
events,
Shuttle
x’
x
Measure
ment events-
26-100
Figure 6: Measurement in the Shuttle frame.
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�
�
with slope 1/u and crosses through the point (L/ 1 − u2 /c2 , uL/ 1 − u2 /c2 ),
so the world
line for the right hand end in the 26-100 coordinates is t =
�
x/u − L 1 − u2 /c2 /u. If we in the 26-100 frame measure the length of the
�
2×4 at t = 0, we find the length to be L 1 − u2 /c2 , which is shorter by a
�
factor 1 − u2 /c2 than the length in the Shuttle frame, in which the 2×4 is
at rest.
Some numbers: the Shuttle orbits the earth every 90 minutes at a distance
of 148 million km, which gives an average velocity of u = 7 × 105 cm/s. This
gives a contraction of L(1−2×10−10 ), so a 1 meter long 2×4 would appear in
the 26-100 frame to be only shorter, about 5 atomic radii! The point here is
that c = 3 × 1010 cm/s is a very large compared with velocities we experience
in everyday life. You should be sure you can carry through this argument for
a 2×4 at rest in the 26-100 frame and measured by the Shuttle astronauts.
Time Dilation
Suppose now the Shuttle commander has been approved to take a 1 hour
nap by Mission Control. He sets his alarm clock (situated at x = 0) to go
off at t = 0 and goes to sleep. When the clock goes off, Mission Control
approves 5 minutes additional rest (the commander being worn out from
2×4 measurements), so he hits the snooze button. Then at tsnooze =300 s, the
alarm goes off again. The situation in the Shuttle frame is shown in Fig. 7.
As before, the 26-100 coordinates and the Shuttle coordinates x = x = 0
correspond at t = t = 0. We do we see in 26-100? Since the coordinates correspond at t = t = 0, in 26-100 we see the first alarm at (0, 0). Applying the
Lorentz Transformation to the�coordinates of the�second alarm in the Shuttle rest frame gives (utsnooze / 1 − u2 /c2 , tsnooze / 1 − u2 /c2 ). Viewed from
�
our frame, this says the commander slept longer by a factor 1/ 1 − u2/c2 of
than he did in the Shuttle rest frame. In the 26-100 frame, the alarm clock
describes a trajectory along and our diagram is shown in Fig. 8.
This is the Lorentz time dilation. Putting in the same numbers as before,
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t’
World line
of alarm clock
Second alarm (0,t’snooze)
First alarm (0,0)
x’
Figure 7: Events of the snooze interval in the Shuttle frame.
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t
ta
t’
Slope=1/u
Second
alarm
x’
x
First
alarm
Figure 8: Alarm clock snooze in the 26-100 frame.
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we find the Shuttle commander appears to get about 1 nanosecond (10−9 s)
more sleep when viewed from the 26-100 frame.
Addition of velocities
Suppose an object moves with velocity v along the x axis of the Shuttle
frame. What is its velocity in the 26-100 frame? With the Galilean transformation (which we know is not correct), we would say the velocity in the
26-100 frame would be v = v + u. If v = c, then v = c + u > c , which contradicts Einstein’s postulate that c is the same in all inertial frames. How do
we add velocities using the Lorentz transformation? Take the total derivative
of each of the equations:
1
1
(x + ut ) → dx = �
(dx + udt )
x= �
2
2
2
2
1 − u /c
1 − u /c
ux
1
udx
�
(t + 2 ) → dt =
(dt + 2 )
c
c
1 − u2 /c2
1 − u2 /c2
t�
1
then
dx
dx + udt
(dx + udt )/dt
= udx� =
�
dt
dt + c2
(dt + udx
)/dt
c2
v + u
=
� .
1 + uv
c2
v =
Notice that for u, v << c, we recover the expression from the Galilean transformation. If v = c, then
v=
c+u
c+u
=c
uc = 1
1 + c2
(c + u)
c
which shows the Lorentz Transformation has the required property: The
velocity of light, c, is the same in all inertial frames.
Finally, suppose an object moves in the S frame with velocity v = vx xˆ + vy ŷ.
We already know how to find vx from the previous paragraph. What about
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vy ?
dy dy
=
dt
γdt + γu
dx
c2
vy
dy/dt
.
=
=
udx/dt
x
)
γ(1 + uv
γ(1 + c2 )
c2
vy =
�
Here we have introduced some new notation: γ = 1/ 1 − u2 /c2 . Also, since
√
u/c appears frequently, pros usually write β = u/c and γ = 1/ 1 − β 2 . If
there are
� several different velocities in the problem, one writes βv = v/c,
γv = 1/ 1 − βv2 , etc.
Invariant interval
The Lorentz transformations 9 amount to a change of coordinates in Minkowski
space; the space of position and time (x, t) (also referred to as spacetime).
In Euclidean space, we can rotate the coordinate system in some way, yet
the distance between to points d = |x1 − x2 | remains the same. Similarly,
the Lorentz transformations leave the interval between two events inchanged.
The interval between two events is just
�
s=
c2 (t1 − t2 )2 − (x1 − x2 )2 .
(13)
We can prove the invariance of the interval in the following way: start with
Eq. 13 measured in S. Then in S , the interval will be
s2 = c2 (t1 − t2 ) − (x1 − x2 )2 − (y1 − y2 )2 − (z1 − z2 )2 .
Now substitute the Lorentz transformation Eq. 9
s2 = c2 ((γt1 + γβx1 /c) − (γt2 + γβx2 /c))2
−((γx1 + γβct1 ) − (γx2 + γβct2 ))2
−(y1 − y2 )2 − (z1 − z2 )2
and expanding and collecting terms gives
s2 = c2 (t1 − t2 )2 − (x1 − x2 )2 − (y1 − y2 )2 − (z1 − z2 )2 = s.
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So s, the interval between two events, is the same in all inertial frames. In
spacetime, s fills the same role as distance does in Euclidian space: they are
both invariant quantities under coordinate transformations.
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